Szabo_hint_chap1.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "init_printing()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Exercise 1.18 hint まず $$|\\tilde \\Phi \\rangle = N \\exp(-\\alpha x^2)$$ を規格化する"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = symbols('x')\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "α = symbols('α')\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{\\sqrt{2} \\sqrt{\\pi}}{2 \\sqrt{α}}$$"
      ],
      "text/plain": [
       "√2⋅√π\n",
       "─────\n",
       " 2⋅√α"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A=simplify(integrate( exp(-α * x ** 2) * exp(-α * x ** 2), (x, -oo, oo)) )\n",
    "A.args[0][0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " 次に、$$   \\langle \\tilde \\Phi | -\\frac{1}{2} \\frac{d^2}{dx^2}|\\tilde \\Phi \\rangle $$ を求める"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- \\frac{\\left(4 x^{2} α^{2} e^{- x^{2} α} - 2 α e^{- x^{2} α}\\right) e^{- x^{2} α}}{2}$$"
      ],
      "text/plain": [
       " ⎛           2            2  ⎞    2   \n",
       " ⎜   2  2  -x ⋅α        -x ⋅α⎟  -x ⋅α \n",
       "-⎝4⋅x ⋅α ⋅ℯ      - 2⋅α⋅ℯ     ⎠⋅ℯ      \n",
       "──────────────────────────────────────\n",
       "                  2                   "
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A=exp(-α * x**2 ) * Rational(-1/2) * diff (diff(exp(-α * x**2 ),x),x)\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{\\sqrt{2} \\sqrt{\\pi} \\sqrt{α}}{4}$$"
      ],
      "text/plain": [
       "√2⋅√π⋅√α\n",
       "────────\n",
       "   4    "
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "AA=integrate(exp(-α * x**2 ) * Rational(-1/2) * diff (diff(exp(-α * x**2 ),x),x), (x, -oo, oo) )\n",
    "AA.args[0][0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "最後に$$ \\langle \\tilde \\Phi | \\delta(x) |\\tilde \\Phi \\rangle $$ を求める"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAOBAMAAADgeEClAAAAIVBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADdcGRXAAAACnRSTlMAMt0Qq5nNdrvvxbMB0AAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABtJREFUCB1jYGBUZmAwCVvMwMDAThbB0rnIHQBAqQoWQ8kakgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$1$$"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "BB=integrate(exp(-α * x**2 ) * DiracDelta(x) * exp(-α * x**2), (x, -oo, oo) )\n",
    "BB"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Exercise 1.19 hint\n",
    " まず$$   \\langle \\tilde \\Phi | -\\frac{1}{2} \\nabla^2 |\\tilde \\Phi \\rangle $$ を求め、つぎに\n",
    " $$   \\langle \\tilde \\Phi | -\\frac{1}{r} |\\tilde \\Phi \\rangle $$ に求める。三次元球面座標になってることに注意。$$ dx dy dx = r^2 \\sin\\theta dr d\\theta d\\phi$$　になる。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "r = symbols('r')\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "π = symbols('π')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- 4 r^{2} α π \\left(2 r^{2} α - 3\\right) e^{- 2 r^{2} α}$$"
      ],
      "text/plain": [
       "                            2  \n",
       "    2     ⎛   2      ⎞  -2⋅r ⋅α\n",
       "-4⋅r ⋅α⋅π⋅⎝2⋅r ⋅α - 3⎠⋅ℯ       "
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B= 4 * π * r ** 2 * exp(-α * r**2 ) * -1/2 * simplify( 1 / r**2 * diff ( r**2 * diff(exp(-α * r**2 ),r),r) )\n",
    "B"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{3 \\sqrt{2} \\sqrt{\\pi} π}{8 \\sqrt{α}}$$"
      ],
      "text/plain": [
       "3⋅√2⋅√π⋅π\n",
       "─────────\n",
       "   8⋅√α  "
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "BB=integrate( B, (r, 0, oo))\n",
    "BB.args[0][0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGwAAAAbCAYAAACZQWq0AAAABHNCSVQICAgIfAhkiAAAA8ZJREFUaIHt2XmoVVUUx/GP6UtLISuJxCywgSbRNDXLbAIJqj/KKIokrLBoQKIBNIrbAP4hkkQYEUEhkk2YSEgU0mT+0aQvmxOiopmojFc2vf5Y+/KOx3vuvecN+sjzhcO957fu2medu87ea+99qNhdjMfL+ABbcNEejaaiJWMxKX0/BF9i/z0XTkVZOnF4Wad9BiCQitacjA7RyyoGmEV4E7/iB6zDiSX8DxZ17NT+D62iES9gvkjSRKzBtziohV8HhuNVzBvIACuaMwr/4IKMdhi6cblI0A5ciSdQa9LWNGzA7/gUs3BJaqOinxgrknNaRjs/ae/gXByJ2fgXmzPHxIzPDJGoRThaJHeDWAKcMaB3kGFeCrwb1+yui+5mnsS7GJrR7kAXJpRoZyNWZ87PEz33xb4G2C7j8TO2G9wJq+l5qIqOMwt8l+IbHJXTn8ZTJWIY1+A6c5J2Sol2es0QvIRt4qYGc8LG4NgWR6MF7jJ8h+Mb2D7B9SViqCcne52F2FqijT6xUIzXs/U8wUUJOyvZl2I61uKnpN2g9dNfPybn2p2L9fgRf4oivtjOQ1dveUBxskaKoWxmifbmJp8R6fwAMfN8uw8xts1xonjen85rmifslmRfL2ZTa7AEq0SxXZHsW1Nb9WN10l/DXXo2AYaKgt0tkvQQluPjpD3ex/tbIdZgZ+PQzDEq2WeKP39kiTYnpNjuFT16nUjWL3YdbvuVYXhL/Dn7Ja2mecJWJft2jcfrC5P9vpx+e9KvzukPJn1JiqdOhyjs3Rr3jHYp6uG1ZL8OH/Wi3cViIf6H+E9G4w1RIweMe+w6HNQ0T9iHyX5Vgf3uZL84p9d72NSMNkMMxc8VtLUg+cwvsA96huXOP8cRJfxX4Yr0fbp4SpZhU5v+I3EMvsdjBb+p16YtOX0K/rZzYb5JTHi6NF6g1reQhrQZ36Ajn7Btoku2y9eZdlaK2dGdJfwnidrzvOgZjZiM3/BZRhslxvb3Rd2rMyd9Xtbiul+UiPF/yWjtz+aWZ/xuTNq1Be0emOwbc/rpSV+Z0UYk7ZU+3MegJ9/DessOPFpgm4KT8LqYjGzK2YiJSiOaDYfE9k6d+jA3plWwFc2pKZ50bBbJ3rfA9+bkuyCnP5L0OTl9S9KLXr/P0j/rsD1Gf/Ww3jBcTK87xcK2EUU9rP4q4wTxbumrdH6bqIfPip2WTlEjx4nZZIdevOXd26hp3MOmJv3hJr6bxTIhvzV0qdjB6MKtOds0PCN2Cv5Kv3svXeec0tFXVFRUVFRUVOyN/Ae04P32KPigiQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$- 4 r π e^{- 2 r^{2} α}$$"
      ],
      "text/plain": [
       "            2  \n",
       "        -2⋅r ⋅α\n",
       "-4⋅r⋅π⋅ℯ       "
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "BBB= 4 * π * r ** 2 * exp(-α * r**2 ) * -1/r* exp(-α * r**2 )\n",
    "BBB"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- \\frac{π}{α}$$"
      ],
      "text/plain": [
       "-π \n",
       "───\n",
       " α "
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "BBBB=integrate( BBB, (r, 0, oo))\n",
    "BBBB.args[0][0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "規格化因子$$   \\langle \\tilde \\Phi |\\tilde \\Phi \\rangle / N^2 $$ も求めておこう"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{\\sqrt{2} \\sqrt{\\pi} π}{4 α^{\\frac{3}{2}}}$$"
      ],
      "text/plain": [
       "√2⋅√π⋅π\n",
       "───────\n",
       "    3/2\n",
       " 4⋅α   "
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C=integrate(4 * π * r** 2 * exp(-α * r**2 )  * exp(-α * r**2), (r, 0, oo) )\n",
    "C.args[0][0]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Exercise 1.20 hint とにかく計算を楽にするのみ。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "θ = symbols('θ')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "c = Matrix([\n",
    "    [cos(θ)],[sin(θ)]\n",
    "])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "O11, O12, O22 = symbols('O11 O12 O22')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "O = Matrix([\n",
    "    [O11, O12], [O12, O22]\n",
    "])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGMAAAAzBAMAAABxm6tRAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEN2ZRGZ2iavNMrvvIlS7SONqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABu0lEQVRIDe2WMUvDQBTHXxpKYm1sM9jFwQ46iyC49iP4EZycBAsO6uTgopuzIoijINTVyQ/gB/BjKFVBQeK9e7l317w7aHRx6MGR6/u/Xy5Jw49Anucw/YjyvAtrxXh6AuaLYgQrNQBsbf4B6W0ebgW2q0Rml+QC4MyPVKMSiS5Ve3voY0RUInN91Z2++BARlcjxQHUnXz5ERIRkn9gcv3kQGRGSfGBz49uDyIiQhr6k5UdYPAc9LetG0cFp1/yVnVfs2Ru2dvqA0xluFEOssnIXfFbRE8CCKuG0o+FEnUGmLoeQFHdJ1U+JuFGza5HWu0LwaUpkIoKkb3aB3WtYulKYRCYiuOHbh9bJ/oYifIgbZfeqh+4Fu/Xw7OJGz3BXE8kebterSHY03sZpzuwcsdwuCvXIKxfm9ASXM2SmPhB+s6+LiMoXRvjNIiIiRPqNERkRIv3GiIwIkX5jxESovZD6yG+MGPVhOaQ+8hsjRn1YDqmP/MaIUR+Wg+rTfmOE1afLAfVpvzHC6tPlgPq03yxi1IflkPq03yxiVlgOqI/8Zhr5iOWQ+shv3GoWWP7/6lut+QnXG/3iE+4HFmvLPXPXWv8AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}O_{11} & O_{12}\\\\O_{12} & O_{22}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡O₁₁  O₁₂⎤\n",
       "⎢        ⎥\n",
       "⎣O₁₂  O₂₂⎦"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "O"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}\\cos{\\left (θ \\right )}\\\\\\sin{\\left (θ \\right )}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡cos(θ)⎤\n",
       "⎢      ⎥\n",
       "⎣sin(θ)⎦"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}\\cos{\\left (θ \\right )} & \\sin{\\left (θ \\right )}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "[cos(θ)  sin(θ)]"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c.T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}\\left(O_{11} \\cos{\\left (θ \\right )} + O_{12} \\sin{\\left (θ \\right )}\\right) \\cos{\\left (θ \\right )} + \\left(O_{12} \\cos{\\left (θ \\right )} + O_{22} \\sin{\\left (θ \\right )}\\right) \\sin{\\left (θ \\right )}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "[(O₁₁⋅cos(θ) + O₁₂⋅sin(θ))⋅cos(θ) + (O₁₂⋅cos(θ) + O₂₂⋅sin(θ))⋅sin(θ)]"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c.T * O * c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}\\left(- O_{11} \\sin{\\left (θ \\right )} + O_{12} \\cos{\\left (θ \\right )}\\right) \\cos{\\left (θ \\right )} - \\left(O_{11} \\cos{\\left (θ \\right )} + O_{12} \\sin{\\left (θ \\right )}\\right) \\sin{\\left (θ \\right )} + \\left(- O_{12} \\sin{\\left (θ \\right )} + O_{22} \\cos{\\left (θ \\right )}\\right) \\sin{\\left (θ \\right )} + \\left(O_{12} \\cos{\\left (θ \\right )} + O_{22} \\sin{\\left (θ \\right )}\\right) \\cos{\\left (θ \\right )}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "[(-O₁₁⋅sin(θ) + O₁₂⋅cos(θ))⋅cos(θ) - (O₁₁⋅cos(θ) + O₁₂⋅sin(θ))⋅sin(θ) + (-O₁₂⋅\n",
       "sin(θ) + O₂₂⋅cos(θ))⋅sin(θ) + (O₁₂⋅cos(θ) + O₂₂⋅sin(θ))⋅cos(θ)]"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(c.T*O*c, θ)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}\\left(- O_{11} \\sin{\\left (θ \\right )} + O_{12} \\cos{\\left (θ \\right )}\\right) \\cos{\\left (θ \\right )} - \\left(O_{11} \\cos{\\left (θ \\right )} + O_{12} \\sin{\\left (θ \\right )}\\right) \\sin{\\left (θ \\right )} + \\left(- O_{12} \\sin{\\left (θ \\right )} + O_{22} \\cos{\\left (θ \\right )}\\right) \\sin{\\left (θ \\right )} + \\left(O_{12} \\cos{\\left (θ \\right )} + O_{22} \\sin{\\left (θ \\right )}\\right) \\cos{\\left (θ \\right )}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "[(-O₁₁⋅sin(θ) + O₁₂⋅cos(θ))⋅cos(θ) - (O₁₁⋅cos(θ) + O₁₂⋅sin(θ))⋅sin(θ) + (-O₁₂⋅\n",
       "sin(θ) + O₂₂⋅cos(θ))⋅sin(θ) + (O₁₂⋅cos(θ) + O₂₂⋅sin(θ))⋅cos(θ)]"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diff(c.T*O*c, θ)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}- O_{11} \\sin{\\left (2 θ \\right )} + 2 O_{12} \\cos{\\left (2 θ \\right )} + O_{22} \\sin{\\left (2 θ \\right )}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "[-O₁₁⋅sin(2⋅θ) + 2⋅O₁₂⋅cos(2⋅θ) + O₂₂⋅sin(2⋅θ)]"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(diff(c.T*O*c, θ))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Exercise 1.22 Hamiltonian行列を作成し、永年方程式を作り、対角化する。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "π = symbols('π')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "F = symbols('F')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "r = symbols ('r')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "ϕ = symbols ('ϕ')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{e^{- r}}{\\sqrt{π}}$$"
      ],
      "text/plain": [
       " -r\n",
       "ℯ  \n",
       "───\n",
       " √π"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ψ_1s=1/sqrt(π) * exp(-r)\n",
    "ψ_1s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{\\sqrt{2} r e^{- \\frac{r}{2}} \\cos{\\left (θ \\right )}}{8 \\sqrt{π}}$$"
      ],
      "text/plain": [
       "      -r        \n",
       "      ───       \n",
       "       2        \n",
       "√2⋅r⋅ℯ   ⋅cos(θ)\n",
       "────────────────\n",
       "      8⋅√π      "
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ψ_2p = 1/sqrt(32 * π) * r * exp(-r/2) * cos(θ)\n",
    "ψ_2p"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "まず、$$   \\langle 1s |  H_0 + F\\cos(\\theta) | 1s \\rangle  = -\\frac{1}{2}$$ 対称性より外場部分は0となる"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "実際、$$   \\langle 1s |   F\\cos(\\theta) | 1s \\rangle  $$を計算すると"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{3 F \\sin^{2}{\\left (π \\right )}}{8}$$"
      ],
      "text/plain": [
       "       2   \n",
       "3⋅F⋅sin (π)\n",
       "───────────\n",
       "     8     "
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "integrate( ψ_1s * F * r * cos(θ) * ψ_1s * r**2 * sin(θ), (r,0,oo) , (θ,0,π), (ϕ, 0, 2 * π))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "次に$$   \\langle 2p |  H_0 + F\\cos(\\theta) | 2p  \\rangle  = -\\frac{1}{8}$$ 同じく対称性より外場部分は0となる。実際"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{15 F \\left(- \\cos^{4}{\\left (π \\right )} + 1\\right)}{8}$$"
      ],
      "text/plain": [
       "     ⎛     4       ⎞\n",
       "15⋅F⋅⎝- cos (π) + 1⎠\n",
       "────────────────────\n",
       "         8          "
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(integrate( ψ_2p * F * r * cos(θ) * ψ_2p * r**2 * sin(θ), (r,0,oo) , (θ,0,π), (ϕ, 0, 2 * π)) )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "最後に$$   \\langle 1s |  H_0 + F\\cos(\\theta) | 2p  \\rangle $$を計算する"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{64 \\sqrt{2} F \\left(- \\cos^{3}{\\left (π \\right )} + 1\\right)}{243}$$"
      ],
      "text/plain": [
       "        ⎛     3       ⎞\n",
       "64⋅√2⋅F⋅⎝- cos (π) + 1⎠\n",
       "───────────────────────\n",
       "          243          "
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(integrate( ψ_1s * F * r * cos(θ) * ψ_2p * r**2 * sin(θ), (r,0,oo) , (θ,0,π), (ϕ, 0, 2 * π)) )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$-cos^3(\\pi) + 1 = 2$$だから"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{128 \\sqrt{2} F}{243}$$"
      ],
      "text/plain": [
       "128⋅√2⋅F\n",
       "────────\n",
       "  243   "
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "128 * sqrt(2) / 243 * F"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "となる。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [],
   "source": []
  },
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